## Sequences with Low-Discrepancy Blue-Noise 2-D Projections

1Université de Lyon, CNRS, LIRIS, France 2Stanford, USA 3Google, USA

#### In Computer Graphics Forum (Proceedings of Eurographics), 2018

Left: our staged per-tile optimized scrambling, applied to a Sobol sequence of sampling points, produces a power spectrum close to Blue Noise. Right: Rendering of a challenging scene featuring depth of field and high specularity (jewels). The sampling was done with the Sobol sequence (top) and our sampler (bottom); both use 256 samples per pixel and 3 light bounces. Note the improvement in aliasing when using our method in comparison to the original Sobol sequence.

### Abstract

Distributions of samples play a very important role in rendering, affecting variance, bias and aliasing in Monte-Carlo and Quasi-Monte Carlo evaluation of the rendering equation. In this paper, we propose an original sampler which inherits many important features of classical low-discrepancy sequences (LDS): a high degree of uniformity of the achieved distribution of samples, computational efficiency and progressive sampling capability. At the same time, we purposely tailor our sampler in order to improve its spectral characteristics, which in turn play a crucial role in variance reduction, anti-aliasing and improving visual appearance of rendering. Our sampler can efficiently generate sequences of multidimensional points, whose power spectra approach so-called Blue-Noise (BN) spectral property while preserving low discrepancy (LD) in certain 2-D projections. In our tile-based approach, we perform permutations on subsets of the original Sobol LDS. In a large space of all possible permutations, we select those which better approach the target BN property, using pair-correlation statistics. We pre-calculate such “good” permutations for each possible Sobol pattern, and store them in a lookup table efficiently accessible in runtime. We provide a complete and rigorous proof that such permutations preserve dyadic partitioning and thus the LDS properties of the point set in 2-D projections. Our construction is computationally efficient, has a relatively low memory footprint and supports adaptive sampling. We validate our method by performing spectral/discrepancy/aliasing analysis of the achieved distributions, and provide variance analysis for several target integrands of theoretical and practical interest.

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### Reference

Hélène Perrier, David Coeurjolly, Feng Xie, Matt Pharr, Pat Hanrahan, Victor Ostromoukhov. Sequences with Low-Discrepancy Blue-Noise 2-D Projections. Computer Graphics Forum (Proceedings of Eurographics), 37(2):339–353, 2018.

@article{perrier18eg,
author = "Perrier, H\'el\ene and Coeurjolly, David and Xie, Feng and Pharr, Matt and Hanrahan, Pat and Ostromoukhov, Victor",
title = "Sequences with Low-Discrepancy Blue-Noise 2-D Projections",
journal = "Computer Graphics Forum (Proceedings of Eurographics)",
year = "2018",
volume = "37",
number = "2",
pages = "339–353",
abstract = "Distributions of samples play a very important role in rendering, affecting variance, bias and aliasing in Monte-Carlo and Quasi-Monte Carlo evaluation of the rendering equation. In this paper, we propose an original sampler which inherits many important features of classical low-discrepancy sequences (LDS): a high degree of uniformity of the achieved distribution of samples, computational efficiency and progressive sampling capability. At the same time, we purposely tailor our sampler in order to improve its spectral characteristics, which in turn play a crucial role in variance reduction, anti-aliasing and improving visual appearance of rendering. Our sampler can efficiently generate sequences of multidimensional points, whose power spectra approach so-called Blue-Noise (BN) spectral property while preserving low discrepancy (LD) in certain 2-D projections. In our tile-based approach, we perform permutations on subsets of the original Sobol LDS. In a large space of all possible permutations, we select those which better approach the target BN property, using pair-correlation statistics. We pre-calculate such “good” permutations for each possible Sobol pattern, and store them in a lookup table efficiently accessible in runtime. We provide a complete and rigorous proof that such permutations preserve dyadic partitioning and thus the LDS properties of the point set in 2-D projections. Our construction is computationally efficient, has a relatively low memory footprint and supports adaptive sampling. We validate our method by performing spectral/discrepancy/aliasing analysis of the achieved distributions, and provide variance analysis for several target integrands of theoretical and practical interest."
}
`

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